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Translating Patterns Into Math Rules

This lesson shows how to translate a visual or numeric pattern into an algebraic rule using a variable for the term position, then use that rule to check and predict terms.

What Does It Mean to Translate a Pattern?

Translating a pattern means taking a sequence of numbers or shapes and rewriting it as a math rule, usually an algebraic expression that uses a variable such as \( n \) for the position of a term. Instead of describing a pattern in words like "start at 1 and add 3 each time," translation turns that description into something you can compute with, such as \( y = 3n - 2 \).

This step comes after you already know how to spot a pattern (see identifying patterns) and how to continue one further along the sequence (see extending patterns). Translation is the bridge between "I can see what is happening" and "I can write down exactly what is happening" using symbols.

From Words to a Math Rule: The Steps

To translate a pattern into a rule, follow the same process every time.

List the terms in order and number their positions starting at \( n = 1 \).
Find the constant amount added (or multiplied) from one term to the next.
Write a starting expression using that constant times \( n \).
Adjust the expression with a plus or minus so it matches the first term exactly.
Check the rule against at least two more terms before trusting it.

The rule \( y = 3n - 2 \) plotted against the term position \( n \), with the first four terms marked.

Worked Example: Translating a Number Pattern

Consider the sequence 1, 4, 7, 10, 13, …

Step 1: the positions are \( n = 1, 2, 3, 4, 5 \) and the values are 1, 4, 7, 10, 13.

Step 2: each term increases by 3, so the pattern is growing at a constant rate of 3 per position.

Step 3: start with \( 3n \), which gives 3, 6, 9, 12, 15 for \( n = 1, 2, 3, 4, 5 \).

Step 4: compare 3 to the actual first term of 1. Since \( 3 - 2 = 1 \), subtract 2 from the expression to get \( y = 3n - 2 \).

Step 5: check \( n = 3 \): \( 3(3) - 2 = 7 \), which matches the third term. Check \( n = 5 \): \( 3(5) - 2 = 13 \), which also matches. The rule \( y = 3n - 2 \) is confirmed.

Position \( n \)	1	2	3	4	5
Value \( y \)	1	4	7	10	13

Worked Example: Translating a Growing Shape Pattern

Suppose a pattern of squares grows like this: 1 square in figure 1, 5 squares in figure 2, 9 squares in figure 3, and 13 squares in figure 4. This is the kind of visual sequence you might have first built while creating growing patterns.

The number of squares increases by 4 each time, so the rule starts as \( 4n \). At \( n = 1 \), \( 4n = 4 \), but the actual value is 1, so subtract 3: \( y = 4n - 3 \).

Check: at \( n = 3 \), \( 4(3) - 3 = 9 \). At \( n = 4 \), \( 4(4) - 3 = 13 \). Both match, so the shape pattern has been fully translated into the algebraic rule \( y = 4n - 3 \).

Reading the Translation Diagram

The diagram below shows the same idea broken into its three parts: the written description, the table of values, and the resulting rule.

Checking a Translated Rule

A rule is only useful if it works for every term, not just the ones used to build it. Always test the rule against at least two terms that were not used while translating, and if it fails, recheck the constant difference between terms since a wrong difference is the most common cause of an incorrect rule.

Once a rule checks out, it can be used to find far away terms instantly. For the rule \( y = 3n - 2 \), the 20th term is simply \( 3(20) - 2 = 58 \), no need to write out twenty terms by hand.

Common Mistakes When Translating Patterns

Using the wrong starting position, such as beginning at \( n = 0 \) instead of \( n = 1 \), which shifts every value in the rule.
Forgetting the adjustment step, and stopping at \( y = 3n \) instead of the correct \( y = 3n - 2 \).
Assuming a constant difference when the pattern actually grows by multiplication rather than addition.
Not verifying the rule against a term beyond the ones already known.

Related lessons

Math 1